Aim of the course is to introduce the basic tools of pseudodifferential calculus, and apply them to analyze the long time dynamics of linear, time dependent Schrödinger equations. A particular emphasis will be given to the problem of growth of Sobolev norms.

**Course contents:**

Part 1: Pseudodifferential operators

- Symbolic calculus: composition, adjoints and quantizations
- Continuity in $L^2$ : the Calderon-Vaillancourt theorem, Garding inequality
- Flow generation and Egorov theorem
- Functional calculus
- Pseudodifferential operators on a manifold and global quantization

Part 2: Applications

- Semiclassical normal form
- Asymptotics of eigenvalues in special cases
- Upper and lower bounds on the growth of Sobolev norms in linear, time depedent Schrödinger equations

**References**

[1] S. Alinhac and P. Gérard, *Pseudo-differential Operators and the Nash-Moser Theorem* (AMS, Graduate Studies in Mathematics, vol. 82, 2007).

[2] X. Saint Raymond, *Elementary Introduction to the Theory of Pseudodifferential Operators* (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.)

[3] M. M. Wong, *An Introduction to Pseudo-differential Operators* (World Scientific, Singapore, 2nd ed., 1999.)

[4] D. Robert,* Autour de l’Approximation Semi-Classique* (Boston etc., Birkhäuser 1987).

[5] M. Taylor, *Pseudo Differential Operators* (Princeton Univ. Press, Princeton, N.J., 1981)

[6] A. Maspero, D. Robert: *On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms*. J. Funct. Anal., 273(2):721–781, 2017.

[7] D. Bambusi, B. Grebert, A. Maspero, D. Robert: *Growth of Sobolev norms for abstract linear Schrödinger Equations*. J. Eur. Math. Soc. (JEMS), in press.

[8] A. Maspero: *Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations*. Math. Res. Lett, in press 2018.

[9] A. Weinstein:* Asymptotics of eigenvalue clusters for the Laplacian plus a potential*. Duke Math. J. 44 (1977), no. 4, 883–892